Continued fractions and semigroups of Möbius transformations
Transcript of Continued fractions and semigroups of Möbius transformations
Continued fractions and semigroups of
Möbius transformations
Ian Short
Wednesday 8 October 2014
Work in preparation
Matthew Jacques
Mairi Walker
A problem about continued fractions
Problem � version I. Determine those �nite sets of real
numbers X with the property that each continued fraction
1
b1 +1
b2 +1
b3 + � � �
with coe�cients in X converges.
Examples
Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.
Examples. X � N 4
X � R+ 4
X = f�1; 1g 8
X = f0g or X = fig 8
�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction
a1
b1 +a2
b2 +a3
b3 + � � �
converges.
Examples
Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.
Examples. X � N 4
X � R+ 4
X = f�1; 1g 8
X = f0g or X = fig 8
�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction
a1
b1 +a2
b2 +a3
b3 + � � �
converges.
Examples
Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.
Examples. X � N 4
X � R+ 4
X = f�1; 1g 8
X = f0g or X = fig 8
�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction
a1
b1 +a2
b2 +a3
b3 + � � �
converges.
Examples
Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.
Examples. X � N 4
X � R+ 4
X = f�1; 1g 8
X = f0g or X = fig 8
�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction
a1
b1 +a2
b2 +a3
b3 + � � �
converges.
Examples
Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.
Examples. X � N 4
X � R+ 4
X = f�1; 1g 8
X = f0g or X = fig 8
�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction
a1
b1 +a2
b2 +a3
b3 + � � �
converges.
Examples
Problem � version I. Determine those �nite sets X with the propertythat each continued fraction with coe�cients in X converges.
Examples. X � N 4
X � R+ 4
X = f�1; 1g 8
X = f0g or X = fig 8
�leszy«ski�Pringsheim theorem. If jbn j > 1+ jan j for each positiveinteger n , then the continued fraction
a1
b1 +a2
b2 +a3
b3 + � � �
converges.
Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
Continued fractions and Möbius transformations
Tn = t1 � t2 � � � � � tn
A problem about sequences of Möbius transformations
De�nition. Given a set of (real) Möbius transformations F ,a composition sequence from F is a sequence
Fn = f1 � f2 � � � � � fn ; where fi 2 F :
Problem � version II. Determine those �nite sets of Möbius
transformations F with the property that every composition
sequence from F converges at 0.
A problem about sequences of Möbius transformations
De�nition. Given a set of (real) Möbius transformations F ,a composition sequence from F is a sequence
Fn = f1 � f2 � � � � � fn ; where fi 2 F :
Problem � version II. Determine those �nite sets of Möbius
transformations F with the property that every composition
sequence from F converges at 0.
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
Real Möbius transformations
z 7!az + b
cz + d; a ; b; c; d 2 R; ad � bc 6= 0
Conjugacy classi�cation of Möbius transformations
elliptic
one �xed pointinside
hyperbolic rotation
parabolic
one �xed pointon the boundary
z 7! z + 1
loxodromic
two �xed pointson the boundary
z 7! �z , � > 1
Conjugacy classi�cation of Möbius transformations
elliptic
one �xed pointinside
hyperbolic rotation
parabolic
one �xed pointon the boundary
z 7! z + 1
loxodromic
two �xed pointson the boundary
z 7! �z , � > 1
Conjugacy classi�cation of Möbius transformations
elliptic
one �xed pointinside
hyperbolic rotation
parabolic
one �xed pointon the boundary
z 7! z + 1
loxodromic
two �xed pointson the boundary
z 7! �z , � > 1
Example � the modular group
Problem � version II. Determine those �nite sets of Möbius
transformations F with the property that every composition
sequence from F converges at 0.
Example. 1 1
0 1
! 1 0
1 1
!
f (z ) = z + 1 g(z ) =z
z + 1
� =
�z 7!
az + b
cz + d: a ; b; c; d 2 Z; ad � bc = 1
�
Example � the modular group
Problem � version II. Determine those �nite sets of Möbius
transformations F with the property that every composition
sequence from F converges at 0.
Example. 1 1
0 1
! 1 0
1 1
!
f (z ) = z + 1 g(z ) =z
z + 1
� =
�z 7!
az + b
cz + d: a ; b; c; d 2 Z; ad � bc = 1
�
Example � the modular group
Problem � version II. Determine those �nite sets of Möbius
transformations F with the property that every composition
sequence from F converges at 0.
Example. 1 1
0 1
! 1 0
1 1
!
f (z ) = z + 1 g(z ) =z
z + 1
� =
�z 7!
az + b
cz + d: a ; b; c; d 2 Z; ad � bc = 1
�
Farey graph
Farey graph
Farey graph
Example � pairs of parabolics
Example. 1 �0 1
! 1 0
� 1
!
f (z ) = z + � g(z ) =z
�z + 1
Theorem. Let F = ff ; gg. Then every composition sequence
from F converges if and only if �� =2 (�4; 0].
Example � even-integer continued fractions.
f (z ) = z + 2 g(z ) =z
2z + 1
Example � pairs of parabolics
Example. 1 �0 1
! 1 0
� 1
!
f (z ) = z + � g(z ) =z
�z + 1
Theorem. Let F = ff ; gg. Then every composition sequence
from F converges if and only if �� =2 (�4; 0].
Example � even-integer continued fractions.
f (z ) = z + 2 g(z ) =z
2z + 1
Example � pairs of parabolics
Example. 1 �0 1
! 1 0
� 1
!
f (z ) = z + � g(z ) =z
�z + 1
Theorem. Let F = ff ; gg. Then every composition sequence
from F converges if and only if �� =2 (�4; 0].
Example � even-integer continued fractions.
f (z ) = z + 2 g(z ) =z
2z + 1
Farey tree
Farey tree
Farey tree
Escaping sequences and general convergence
De�nition. A sequence Fn of Möbius transformations is an escaping
sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).
Escaping sequences and general convergence
De�nition. A sequence Fn of Möbius transformations is an escaping
sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).
Escaping sequences and general convergence
De�nition. A sequence Fn of Möbius transformations is an escaping
sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.
An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).
Escaping sequences and general convergence
De�nition. A sequence Fn of Möbius transformations is an escaping
sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.
An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).
Escaping sequences and general convergence
De�nition. A sequence Fn of Möbius transformations is an escaping
sequence if the sequence Fn(�) accumulates only on the unit circle inthe Euclidean metric.An escaping sequence Fn converges generally to a point p on theunit circle if Fn(�)! p as n !1 (in the Euclidean metric).
Classical convergence implies general convergence
Backwards limit sets and conical limit sets
De�nition. The backwards limit set of an escaping sequence Fn isthe set of accumulation points of the sequence F�1
n(�).
Backwards limit sets and conical limit sets
De�nition. The backwards limit set of an escaping sequence Fn isthe set of accumulation points of the sequence F�1
n(�).
Backwards limit sets and conical limit sets
De�nition. The backwards limit set of an escaping sequence Fn isthe set of accumulation points of the sequence F�1
n(�).
Backwards limit sets and conical limit sets
De�nition. The backwards limit set of an escaping sequence Fn isthe set of accumulation points of the sequence F�1
n(�).
Return to the problem
Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.
Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.
Theorem. The conjecture is true if F has order two.
Return to the problem
Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.
Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.
Theorem. The conjecture is true if F has order two.
Return to the problem
Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.
Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.
Theorem. The conjecture is true if F has order two.
Return to the problem
Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.
Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.
Theorem. The conjecture is true if F has order two.
From Möbius transformations to semigroups
Lemma. Let F be a �nite set of Möbius transformations, and
let S be the semigroup generated by F .
The following are
equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
De�nition. A semigroup S is inverse free if no element of S
has an inverse in S . Equivalently, S is inverse free if it does not
contain the identity element.
From Möbius transformations to semigroups
Lemma. Let F be a �nite set of Möbius transformations, and
let S be the semigroup generated by F . The following are
equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
De�nition. A semigroup S is inverse free if no element of S
has an inverse in S . Equivalently, S is inverse free if it does not
contain the identity element.
From Möbius transformations to semigroups
Lemma. Let F be a �nite set of Möbius transformations, and
let S be the semigroup generated by F . The following are
equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
De�nition. A semigroup S is inverse free if no element of S
has an inverse in S .
Equivalently, S is inverse free if it does not
contain the identity element.
From Möbius transformations to semigroups
Lemma. Let F be a �nite set of Möbius transformations, and
let S be the semigroup generated by F . The following are
equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
De�nition. A semigroup S is inverse free if no element of S
has an inverse in S . Equivalently, S is inverse free if it does not
contain the identity element.
Discrete semigroups
Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) each composition sequence from F converges generally
(iii) S is discrete and inverse free.
Problem � version IV. Classify the inverse-free discrete semigroups.
Discrete semigroups
Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F .
Then, with one exception, thefollowing are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) each composition sequence from F converges generally
(iii) S is discrete and inverse free.
Problem � version IV. Classify the inverse-free discrete semigroups.
Discrete semigroups
Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) each composition sequence from F converges generally
(iii) S is discrete and inverse free.
Problem � version IV. Classify the inverse-free discrete semigroups.
Discrete semigroups
Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) each composition sequence from F converges generally
(iii) S is discrete and inverse free.
Problem � version IV. Classify the inverse-free discrete semigroups.
Discrete semigroups
Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) each composition sequence from F converges generally
(iii) S is discrete and inverse free.
Problem � version IV. Classify the inverse-free discrete semigroups.
Discrete semigroups
Lemma. Let F be a �nite set of Möbius transformations, and let S bethe semigroup generated by F . The following are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) the identity element does not belong to the closure of S .
Theorem. Let F be a set of two Möbius transformations, and let S bethe semigroup generated by F . Then, with one exception, thefollowing are equivalent:
(i) each composition sequence from F is an escaping sequence
(ii) each composition sequence from F converges generally
(iii) S is discrete and inverse free.
Problem � version IV. Classify the inverse-free discrete semigroups.
Recap
Problem � version I. Determine those �nite sets of real numbers Xwith the property that each continued fraction with coe�cients in X
converges.
Problem � version II. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges at 0.
Problem � version III. Determine those �nite sets of Möbiustransformations F with the property that every composition sequencefrom F converges generally.
Problem � version IV. Classify the inverse-free discrete semigroups.
Selected literature on Möbius semigroups
Some Moebius semigroups on the 2-sphereC.S. BallantineJ. Math. Mech., 1962
On certain semigroups of hyperbolic isometriesT. Jørgensen and K. SmithDuke Math. J., 1990
Complex dynamics of Möbius semigroupsD. Fried, S.M. Marotta and R. StankewitzErgodic Theory Dynam. Systems, 2012
Entropie des semi-groupes d'isométrie d'un espace hyperboliqueP. MercatTo be published
Exceptional semigroup
f (z ) = 2z g(z ) = 12z + 1
Two-generator Fuchsian groups literature
The classi�cation of discrete 2-generator subgroups of PSL(2; R)J.P. MatelskiIsrael J. Math., 1982
An algorithm for 2-generator Fuchsian groupsJ. Gilman and B. MaskitMichigan Math. J., 1991
Two-generator discrete subgroups of PSL(2;R)J. GilmanMem. Amer. Math. Soc., 1995
Two-generator Fuchsian groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky groups
Schottky semigroups
Schottky semigroups
Schottky semigroups
Schottky semigroups
Schottky semigroups
Schottky semigroups
Schottky semigroups
Reverse triangle inequality
Two-generator inverse-free discrete semigroups
elliptic elliptic
8
elliptic parabolic
8
elliptic loxodromic
8
parabolic parabolic
4
parabolic loxodromic
4
loxodromic loxodromic
4
Two-generator inverse-free discrete semigroups
elliptic elliptic 8
elliptic parabolic 8
elliptic loxodromic 8
parabolic parabolic
4
parabolic loxodromic
4
loxodromic loxodromic
4
Two-generator inverse-free discrete semigroups
elliptic elliptic 8
elliptic parabolic 8
elliptic loxodromic 8
parabolic parabolic 4
parabolic loxodromic 4
loxodromic loxodromic 4
Pairs of parabolics
Pairs of parabolics
�
�
�
�
Pairs of loxodromics
Pairs of loxodromics
�
�
�
�
�
�
Solution to Problem III
Theorem. A set F = ff ; gg of two Möbius transformations has
the property that every composition sequence from F converges
generally if and only if one of the following conditions is
satis�ed:
(i) f and g are parabolic and fg is not elliptic
(ii) one of f or g is loxodromic and the other is either
parabolic or loxodromic, and fgn and f ng are not elliptic
for any positive integer n .
Return to the original problem
Problem � version I. Determine those �nite sets X with the
property that each continued fraction with coe�cients in X
converges.
�1
b + zjbj < 2 �
1
b + zjbj > 2
Theorem. A �nite set X has the property that every minus
continued fraction with coe�cients in X converges if and only
if X � (�1; 2] [ [2;+1).
Return to the original problem
Problem � version I. Determine those �nite sets X with the
property that each minus continued fraction with coe�cients in
X converges.
�1
b + zjbj < 2 �
1
b + zjbj > 2
Theorem. A �nite set X has the property that every minus
continued fraction with coe�cients in X converges if and only
if X � (�1; 2] [ [2;+1).
Return to the original problem
Problem � version I. Determine those �nite sets X with the
property that each minus continued fraction with coe�cients in
X converges.
�1
b + zjbj < 2 �
1
b + zjbj > 2
Theorem. A �nite set X has the property that every minus
continued fraction with coe�cients in X converges if and only
if X � (�1; 2] [ [2;+1).
Return to the original problem
Problem � version I. Determine those �nite sets X with the
property that each minus continued fraction with coe�cients in
X converges.
�1
b + zjbj < 2 �
1
b + zjbj > 2
Theorem. A �nite set X has the property that every minus
continued fraction with coe�cients in X converges if and only
if X � (�1; 2] [ [2;+1).
Return to the original problem
Problem � version I. Determine those �nite sets X with the
property that each continued fraction with coe�cients in X
converges.
b
c
bc 2 (�4; 0]
Theorem. A set fb; cg of real numbers has the property that
every continued fraction with coe�cients in fb; cg converges ifand only if bc =2 (�4; 0].
Return to the original problem
Problem � version I. Determine those �nite sets X with the
property that each continued fraction with coe�cients in X
converges.
b
c
bc 2 (�4; 0]
Theorem. A set fb; cg of real numbers has the property that
every continued fraction with coe�cients in fb; cg converges ifand only if bc =2 (�4; 0].
Return to the original problem
Problem � version I. Determine those �nite sets X with the
property that each continued fraction with coe�cients in X
converges.
b
c
bc 2 (�4; 0]
Theorem. A set fb; cg of real numbers has the property that
every continued fraction with coe�cients in fb; cg converges ifand only if bc =2 (�4; 0].
Examples
1
1+ z
1
2+ z
1
b + zb 2 N 1
b + zb 2 R
1
z
z + 1z
z + 1z + 2
z
�2z + 1
�
�
Examples
1
1+ z
1
2+ z
1
b + zb 2 N
1
b + zb 2 R
1
z
z + 1z
z + 1z + 2
z
�2z + 1
�
�
Examples
1
1+ z
1
2+ z
1
b + zb 2 N 1
b + zb 2 R
1
z
z + 1z
z + 1z + 2
z
�2z + 1
�
�
Examples
1
1+ z
1
2+ z
1
b + zb 2 N 1
b + zb 2 R
1
z
z + 1z
z + 1z + 2
z
�2z + 1
�
�
Examples
1
1+ z
1
2+ z
1
b + zb 2 N 1
b + zb 2 R
1
z
z + 1z
z + 1
z + 2z
�2z + 1
�
�
Examples
1
1+ z
1
2+ z
1
b + zb 2 N 1
b + zb 2 R
1
z
z + 1z
z + 1z + 2
z
�2z + 1
�
�
The unknown
Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.
More generators. Classify those sets of transformations F of sizegreater than two with the property that every composition sequencefrom F converges generally.
Higher dimensions. Classify those sets of complex transformations Fof size two with the property that every composition sequence from F
converges generally.
The unknown
Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.
More generators. Classify those sets of transformations F of sizegreater than two with the property that every composition sequencefrom F converges generally.
Higher dimensions. Classify those sets of complex transformations Fof size two with the property that every composition sequence from F
converges generally.
The unknown
Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.
More generators. Classify those sets of transformations F of sizegreater than two with the property that every composition sequencefrom F converges generally.
Higher dimensions. Classify those sets of complex transformations Fof size two with the property that every composition sequence from F
converges generally.
The unknown
Conjecture. If every composition sequence from F is an escapingsequence, then every composition sequence converges generally.
More generators. Classify those sets of transformations F of sizegreater than two with the property that every composition sequencefrom F converges generally.
Higher dimensions. Classify those sets of complex transformations Fof size two with the property that every composition sequence from F
converges generally.
Gallery1
1Created using lim by Curt McMullen
Gallery1
1Created using lim by Curt McMullen
Gallery1
1Created using lim by Curt McMullen
Gallery1
1Created using lim by Curt McMullen
Thank you for your attention.